2. GROUND RULES
• The same ground rules apply.
• Raise your hand.
There is a little hand in the bottom-left to do
that.
• Everyone raise your hand and then put down
your hand.
• Everyone type something into the message
board
• Let’s still have fun!
3. QUIZ TIME
• What is the definition of calculus? (remember, we talked about it
last time)
• What are the three pieces of a function? Look at the function
machine below
What is this part
What is this part What is this part
4. REVIEW
• A function is like a vending machine.
You put your money in…and most of
the time you get candy out!
• A function works kinda in the same way
5. REVIEW
• The best example is when people like Einstein write math
functions that “model” the universe.
• There is no way we can understand and go out to check the
universe. We can model it with functions
6. VISUAL FUNCTIONS
• What is a coordinate
plane?
• We have an x and a y!
• The x and y always stay
together on a
coordinate plane.
• (x, y) is how you write a
coordinate.
7. VISUAL FUNCTIONS
• Functions are a way to figure out your y!
• This means if I know my x, and a function, then I
can know where to place a dot!
• Go to Desmos
8. FUNCTIONS AND RATES OF CHANGE
Distance and Time
Speed
Function
How fast
you were
going
Input Output
• Example- If I walk 20 miles in 2 hours the
• Where is the calculus in what we were just
talking about?
Speed =
Change in Distance
Change in Time
=
20 miles
2 hours
= 10 miles per hour
What does it mean to change you distance?
9. FUNCTIONS AND RATES OF CHANGE
Distance and Time
Speed
Function
How fast
you were
going
Input Output
• Example- If I walk 20 miles in 2 hours the
•
• Where is the calculus in what we were just talking about?
Speed =
Change in Distance
Change in Time
=
20 miles
2 hours
= 10 miles per hour
10. LET’S CALCULATE THE SPEED OF THE ANDROMEDA GALAXY
• The distance of the Andromeda galaxy from earth is 2.5 million light-
years away
• The time it would take for the Andromeda galaxy to travel this
distance is 4.5 billion years
•
• The average human only walk 110,000 miles in 80 years!!!!
• Remember this is a very rough guess for the speed of the
Andromeda galaxy!!!!!!
Speed =
Change in Distance
Change in Time
=
1,410,000,000,000,000,000,000
4,000,000,000
= 3,525,000,000 miles per year
11. WHAT IS YOUR SPEED CHANGES?
• So what does it mean when your speed
change?
• This is called acceleration.
•
• As you speed up or slow down, your
acceleration goes up or goes down!
• What are some examples of
acceleration? What is one special case of
acceleration because we live on earth?
Acceleration =
Change of Speed
Change in Time
12. DERIVATIVE TIME
• The derivative of a function at a certain point is a measure of the rate
at which that function is changing.
• Remember that Speed and Acceleration are both functions!
• Speed is a derivative of distance
• Acceleration is a derivative of Speed
•
•
d
dt
(Distance) = Speed
d
dt
(Speed) = Acceleration
13. LET’S BREAK THIS DOWN
DERIVATIVES
d
dt
(Distance) = Speed
Take the
derivative
Of
Distance
This gives
you Speed
As your distance changes with time, you get speed
What does it mean to change you distance?
15. FUNCTIONS AND RATES OF CHANGE
Distance and Time
Speed
Function
How fast
you were
going
Input Output
This is a derivative!!!
• Example- If I walk 20 miles in 2 hours the
• Speed =
Change in Distance
Change in Time
=
20 miles
2 hours
= 10 miles per hour
16. WHAT IS YOUR SPEED CHANGES?
• So what does it mean when your
speed change?
• This is called acceleration.
•
• This is a derivative!!!!!
Acceleration =
Change of Speed
Change in Time
17. LET’S REVIEW
• A Derivative of a function is a measure of the rate at which that
function is changing!
•
This means that as my distance changes over time I get speed.
What happens if my distance does not change?
•
This means that as my speed changes over time I get acceleration.
What happens if my speed does not change?
d
dt
(Distance) = ?
d
dt
(Speed) = ?
